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Differential Forms: Theory and Practice

Steven H. Weintraub
Academic Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

Differential forms have been moving into the undergraduate curriculum starting at least as early as Edwards’s Advanced Calculus in 1969 or so, just about the time I was taking advanced calculus myself. There are now at least a few advanced calculus textbooks that introduce differential forms in two and three dimensions as part of their treatment of vector calculus. Steven Weintraub published an earlier book, Differential Forms: A Complement to Vector Calculus, written for the third semester of calculus, with the goal of showing how differential forms can unify the treatment of multivariable calculus. This new book is a further development at a deeper level and with greater rigor.

Much of the book is aimed directly at a rigorous development of the generalized Stokes’s theorem starting from a very basic treatment of differential forms. The author expects his readers to be familiar with basic aspects of point-set topology and linear algebra. A little bit of group theory comes in too, mostly toward the end of the book.

The author first treats differential forms in Rn. He begins with an introductory discussion of Euclidean space, tangent spaces, vector fields and exterior differentiation. He then proceeds to k-forms, orientation and signed volume, Poincaré’s lemma and its converse. Before carrying this machinery forward to smooth manifolds, the author includes an entire chapter on push forwards and pull backs in Rn.

Then, with the tools in place, we move on to smooth manifolds. This part has an extended discussion of orientation — first intuitive and then formal. A short chapter on vector bundles serves to introduce tangent and cotangent bundles. Integration of differential forms comes next; this is done in stages — integrals of 0-forms over points, 1-forms over curves, etc., culminating with integration on chains. A capstone chapter proves the generalized Stokes’s theorem and shows how it leads to the standard Gauss, Green and Stokes theorems. The book then concludes with an introduction to de Rham cohomology.

This is a rigorous and well-written treatment of differential forms with a careful and detailed progression from very basic notions. My main concern is about its fit in the curriculum. The book is situated somewhere between advanced calculus and differential geometry, and it seems that it should belong to a sequence going somewhere — but the destination is not so clear. The final chapter on de Rham cohomology adds to the mystery since the motivation for it would be pretty unclear to a novice.

A couple of moderately annoying things about the book are probably due to the publisher. The font used here is fairly large, which is good, but the text overfills the pages and leaves very small margins. The pages look way overcrowded. Also, the index is poor. Several times I wanted to go back and review something I’d seen, but the index was much too sparse to help.

Bill Satzer ([email protected]) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Differential Forms
The Algrebra of Differential Forms
Exterior Differentiation
The Fundamental Correspondence
Oriented Manifolds
The Notion Of A Manifold (With Boundary)

Differential Forms Revisited
Push-Forwards And Pull-Backs

Integration Of Differential Forms Over Oriented Manifolds
The Integral Of A 0-Form Over A Point (Evaluation)
The Integral Of A 1-Form Over A Curve (Line Integrals)
The Integral Of A2-Form Over A Surface (Flux Integrals)
The Integral Of A 3-Form Over A Solid Body (Volume Integrals)
Integration Via Pull-Backs

The Generalized Stokes' Theorem
Statement Of The Theorem
The Fundamental Theorem Of Calculus And Its Analog For Line Integrals
Green's And Stokes' Theorems
Gauss's Theorem
Proof of the GST

For The Advanced Reader
Differential Forms In IRN And Poincare's Lemma
Manifolds, Tangent Vectors, And Orientations
The Basics of De Rham Cohomology

Answers To Exercises
Subject Index